{ 1,1, 2, 3, 5, 8,13, 21, }

Transcription

1 Chpter 8 Sequeces d Ifiite Series Sectio 8. Overview A ifiite sequece is list of umbers i defiite order,, 3,,, For emple, cosider the ifiite sequece, 4,8,,, the first term =, the secod term = 4, d the th term is =. Note the iteger is clled the ide of the sequece d idictes the positio where occurs i the list. Defiitio A ifiite sequece of umbers is fuctio whose domi is the set of positive itegers. Nottio for sequeces:,,,, List terms: { } Write th term of sequece { } Write th term of sequece with ide { } we choose. = (ote we c strt with y umber Emples: =,,,,, = 3 =,,,,,,! 6 4! { } = {,,,,, ( ), } = 0 Some sequeces do ot hve defiig equtio. For emple, The sequece of the popultio size o J st of ech yer. Deciml vlues of π. {, 4,, 5, 9, } Fibocci sequece { } is defied recursively =, =, = + for 3 {,,, 3, 5, 8,3,, } Emple: Suppose the list of term of sequece is the th term,. 4 8,,,,, write epressio for 3 9 7

2 8. Sequeces From sectio 8.: Defiitio pproch uique umber L s icreses, the we sy lim = L eists, d the sequece coverges to L. If the terms of the sequece do ot pproch sigle umber s icreses, the sequece hs o limit d the sequece diverges. If the terms of sequece { } Theorem 8. Limits of Sequeces from Limits of Fuctios f Suppose f is fuctio such tht = for ll positive itegers. If the lim the the limit of the sequece { } is lso L. Theorem 8. Properties of Limits of Sequeces Suppose tht c is costt d the limits lim d lim b eist. Costt Multiple Rule: lim c = c lim. Sum/Differece Rule: [ ] 3. Product Rule: [ ] 4. Quotiet Rule: lim ± b = lim ± limb lim b = lim lim b lim lim = b lim b provided limb 0 Emples: Determie if the sequece{ } coverges, if it coverges fid the limit = + f = L,. = = + 4. l = l

3 5. = si 6. = + Defiitio A sequece is clled o-decresig if + for ll. A sequece { } is clled oicresig if + for ll. A sequece tht is either o-icresig or o-decresig is sid to be mootoic. Emple: Show = + 3 is o-icresig. Emple: Show = + is o-decresig: Defiitio: A sequece { } is bouded if there eists umber M such tht Theorem 8.5 Bouded Mootoic Sequece A bouded mootoic sequece coverges M for ll. Theorem 8.3 Geometric Sequece Let r be rel umber. The 0 if r < lim r = if r = DNE if r or r > If r > 0, the { r } coverges or diverges mootoiclly. If 0 diverges by oscilltio. Emple: r <, the { } r coverges or 3

4 Theorem 8.4 Squeeze Theorem for Sequeces Let { },{ b}, d { c } be sequeces with b c for ll itegers greter th some N. If lim = limc = L the limb = L. Emple : Fid the limit of si = + Theorem 8.6 Growth Rte of Sequeces The followig sequeces re ordered ccordig to icresig growth rtes s ; tht is, if { } ppers before { b } i the list, the lim =0: b q p p r p + s { l } { } { l } { } { b } {! } { } The orderig pplies for positive rel umbers p, q, r, d s d b >. Emple: Fid the limit of! = Defiitio A sequece { } hs limit L d we write lim = L or L s if we c me the terms s close to L s we lie by tig sufficietly lrge.[formlly, if for every ε > 0, there eists N such tht L < ε wheever > N.] If lim eists, we sy tht the sequece coverges (or is coverget) d coverges to L. Otherwise, we sy tht the sequece diverges. 4

5 Sectio 8.3 Ifiite Series Defiitio A ifiite series is epressio of the form Where { } = = is ifiite sequece of rel umbers. C ifiite series hve sum? Loo t = = Defiitio The th prtil sum of series is s = = i i= So for our emple, s =, s = + =, s 3 = + + =, s4 = =, otice tht the prtil sums re pprochig oe. Thus, we sy tht the series coverges d =. = = Note: The prtil sums of series will form sequece! Some series will diverge, i.e. will ot hve sum. For istce, cosider series, the th prtil sum goes to ifiity s goes to ifiity., it is diverget = Defiitio The series to S. If the sequece { } is sid to coverge with sum S if the sequece of prtil sums { s } = Emple: Is the series ( ) = lim = lim s = S = = s does ot coverge, the series diverges d hs o sum. coverget or diverget? = coverges 5

6 Emple: Telescopig Series Show tht coverges d fid it sum. = + Defiitio: Geometric Series The series is sid to be geometric series if ech term fter the first term is fied = multiple of the term immeditely before it The geometric series Proof: If r =, the diverges. 3 + r + r + r + + r + =r = 0 r with 0 diverges if = 0 S = r = = 0 r r d coverges if r < with sum s = = ±, sice lim s does ot eist, the the geometric series If r we hve s = + r + r + + r Ad rs = r + r + r + + r Subtrctig theses equtios we get s rs = r 3 ( r ) s = r ( r ) If r <, lim s = lim = lim r = r r r r Thus whe r <, the geometric series is coverget with sum r diverget. d if r the series is 3 Emple: Does coverge or diverge? If it coverges, wht is its sum? 7 = 0 6

7 Emple: Does 4 coverge or diverge? If it coverges, wht is its sum? 0 3 = Emple: Does + coverge or diverge? If it coverges, wht is its sum? 5 = Emple: Write 5.43 s rtiol umber. 7

8 Sectio 8.4 The Divergece d Itegrl Test Theorem 8.8 Properties of Coverget Series. Suppose coverges to A d let c be rel umber. The series c coverges d c c ca = =.. Suppose coverges to A d b coverges to B. The series ( ± b) coverges ± b = ± b = A ± B. d 3. Whether series coverges does ot deped o fiite umber of terms dded to or removed from the series. Specificlly, if M is positive iteger, the d = = M both coverge or both diverge. However, the vlue fo the coverget series does chge if the ozero terms re dded or deleted. Emple: Fid the sum of = Theorem 8.9 Divergece Test If coverges, the lim = 0. Equivletly, if lim 0, the the series diverges. Proof: Let s = , so = s s. Sice lim s is coverget the the sequece { s } = = S so lim = lim( s s ) = S S = 0. of the prtil sums is coverget d Emple: Show rct is diverget. = Emple: Show 3+ 5 is diverget. = + Note: If lim = 0, we do ot ow if the series = coverges or diverges. 8

9 Theorem 8. The Itegrl Test Suppose f is cotiuous, positive, decresig fuctio for =,,3. The d f = d d let f = for Either both coverge or both diverge. I the cse of covergece, the vlue of the itegrl is ot, i geerl, equl the vlue of the series. Cosider the followig: i i= f d 3 4 f d i= i 3 (i) If f d is coverget, the i f d f d sice f 0 i= 9

10 Therefore, S = + + f d = M. Sice S i i= is bouded from bove. Also S S + S+ M for ll the the sequece { S } + = sice = f ( + + ) 0 Thus { } S is odecresig bouded from bove sequece so it is coverget. Thus (ii) If f d is diverget, the f d s becuse f 0. But f ( ) d S tht S d so is diverget. = is coverget. = so S which implies Emple: Show e is coverget with the itegrl test. = Emple: Show the Hrmoic Series, is diverget with the itegrl test. = Emple: For wht vlues of p re the p-series is coverget. p = 0

11 Theorem 8. Covergece of the p-series The p-series coverges whe p > d diverges whe p. p = To estimte the sum of series, we dd the first terms together. Now the error of estimtio is the differece betwee the ctul sum, S, d the prtil sum, S. We will cll the differece the remider, R. Now R = S S = We c boud the remider by comprig the sum of the res of rectgles with re uder the y = f for. We see tht curve So R f d d R f d + f d R f d which implies tht S f d S S f d Emple: Estimte the sum of the series usig the iequlity bove d = 0. =

12 Sectio 8.5 The Rtio, Root d Compriso Tests Theorem 8.4 The Rtio Test Let be series with positive terms d suppose tht lim + = ρ The. If ρ <, the series coverges.. If ρ > or ρ is ifiite, the series diverges 3. If ρ =, the test is icoclusive. Commet: This test is very useful if cotis fctoril or c where c is some costt. However, this test will be icoclusive if loos lie p-series. Emple: Determie if the series! is coverget or diverget. 0 = 3 Emple: Determie if the series is coverget or diverget. 3 = Theorem 8.5 The Root Test Let be series with positive terms d suppose tht lim = ρ The. If ρ <, the series coverges.. If ρ > or ρ is ifiite, the series diverges 3. If ρ =, the test is icoclusive. Commet: This test is useful whe it is esy to te the th root of. 3 Emple: Determie if the series = is coverget or diverget.

13 Emple: Determie if the series is coverget or diverget. ( ) = Emple: Determie if the series = is coverget or diverget. Theorem 8.6 Direct Compriso Test Suppose tht d b re series with positive terms. If 0 < b d b is coverget the is coverget.. If b > 0 d b is diverget the is diverget. Emple: Determie if the series is coverget or diverget. 3 = + Emple: Determie if the series is coverget or diverget. = 3

14 Theorem 8.7 Limit Compriso Test Suppose tht d b re series with positive terms d lim = L b. If 0 < L < the d b both coverge or both diverge.. If L = 0 d b coverges the coverges. 3. If L = d b diverges, the diverges. Emple: Determie if the series + 5 is coverget or diverget. 3 7 = + Emple: Determie if the series 5 is coverget or diverget. 3 = + rct Emple: Determie if the series is coverget or diverget.. = Emple: Determie if the series is coverget or diverget. l = + 4

15 Sectio 8.6 Altertig Series, Absolute d Coditiol Covergece A series where the terms lterte betwee positive d egtive is clled ltertig series: Where is positive umber. + or ( ) = = For istce, the ltertig hrmoic series is = + = + = Theorem 8.8 Altertig Series Test The series = = + + Coverges if the followig coditios re stisfied. > 0 > N (i.e. the terms i the series re o-icresig). +. lim = 0 + Emple: Determie if the ltertig hrmoic series is coverget or diverget. = Emple: Determie if the series ( ) is coverget or diverget. + = Emple: Determie if the series π cos = is coverget or diverget. 5

16 Theorem 8.0 Remider i Altertig Series + is the sum of ltertig series tht stisfies the coditios of the = If S = Altertig Series Test the R = S S + Emple: Fid the sum of the series = 0 ( ) correct to three deciml plces.! Defiitio Absolute d Coditiol Covergece Assume the ifiite series coverges. The series coverges bsolutely if the series coverges. Otherwise, the series coverges coditiolly. Emple: Determie if the series diverges. = ( 00) coverges bsolutely, coverges coditiol or! Emple: Determie if the series diverges. = + coverges bsolutely, coverges coditiol or 4 6

17 Theorem 8. Absolute Covergece Implies Covergece If coverges, the coverges. If diverges, the diverges. Emple: Determie if the series si coverges or diverges. = Rerrgig Series If we rerrge the terms of fiite sum, the the vlue of the sum remis uchged. If we rerrge the terms of ifiite series,, we will chge the order of the terms. For istce, we might hve The Rerrgemet Theorem for Absolutely Coverget Series If is bsolutely coverget with sum S the y rerrgemet of hs the sme sum S. However, y coditiolly coverget series c be rerrged to give differet sum. Cosider the ltertig hrmoic series + = = l = Now multiply the series by the + + = l Now isert zeros i betwee the terms of this series, = l Now l + l is = l Notice we hve the sme terms but i differet order d we hve chged the sum of the series. Riem proved tht If is coditiolly coverget series d r is y rel umber whtsoever, the there is rerrgemet of tht hs sum equl to r. 7

18 Summry for Determiig the covergece or divergece of. Loo t the form of the series.. Is it p-series,, the series coverges if p > ; it diverges if p. p b. Is it geometric series r, the series coverges if r < ; it diverges if r. (The sum of coverget geometric series is S = / ( r).) c. Is it telescopig series? If it is fid the th prtil sum, S the if lim S eists, the series coverges d hs sum S = lim S. d. Is it ltertig series, if it is use the Altertig Series Test.. Is it positive term series? If it is,. Ad the terms re similr to those of geometric series or p-series, try the Direct Compriso Test or the Limit Compriso Test. b. Ad if = f d f d is esy to evlute use the Itegrl Test, remember to verify tht the coditios of the itegrl test re stisfied. c. Ad if the terms of the series coti fctorils or c (c is positive costt) try the Rtio Test. d. Ad if it is esy to te the th root of the terms, try the Root Test. 3. If the series hs some egtive terms. Chec if coverges; if it does the coverges sice bsolute covergece implies covergece. 4. If you c esily determie tht lim 0, the the th Term Test for Divergece idictes tht the series diverges. Test the series for covergece or divergece 3.! =. = = + ( 3) 3 8

19 4. = si 5. = 6. l ( ) 3 = + 7. = rct 8. ( ) = l 9. = + ( ) cos / = To determie if series coverges bsolutely, coverges coditiolly or diverges.. Loo t if it coverges the the series coverges bsolutely.. If diverges the chec usig the process bove; if it coverges the the series is coditiolly coverget d if it diverges the the series is diverget. 9

20 Chpter 9 Power Series Sectio 9. Approimtig Fuctios with Polyomils Defiitio: Tylor Polyomil Defiitio Let f be fuctio with derivtives of order for =,,, N i some itervl cotiig s iterior poit. The for y iteger from 0 through N, the Tylor polyomil of order geerted by f t = is the polyomil ( f " ) f P = f + f '( ) + ( ) + + ( )!! Emple: Fid the Tylor series for f order 4. = t = 4. Also fid the Tylor polyomil of Tylor s Theorem If f d its first derivtives f ', f ",, f re cotiuous o the closed itervl betwee d b d f is differetible o the ope itervl betwee d b, the there eists umber c betwee d b such tht ( ) ( + f " ) f f ( c) + f ( b) = f + f '( b ) + ( b ) + + ( b ) + ( b )!! +! Tylor s Formul If f hs derivtes of ll order o ope itervl I cotiig, the for ech positive iteger d for ech i I. ( f " ) f f = f + f '( ) + ( ) + + ( ) + R ( )!! ( + f ) ( c) + Where R = ( ) for some c betwee d b. +! Notice f P R P R P ( ) If R 0 s = + for I is the polyomil pproimtio of order is clled the remider of order or the error term for the pproimtio of f by for ll I, the we sy tht the Tylor series geerted by f t = coverges to f o I. 0

21 The Remider Estimtio Theorem If there is positive costt M such tht the remider term R ( + ) f t M for ll t betwee d, iclusive, the i Tylor s Theorem stisfies the iequlity ( +! ) If this coditio holds for every d the other coditios of Tylor s Theorem re stisfied by f, the the series coverges to f ( ). + R M Emple: Show tht cos is equl to the sum of it Mcluri series. Sectio 9. Properties of Power Series Defiitio A power series bout = 0 is series of the form c = c0 + c + c + + c + = 0 A power series bout = is series of the form c( ) = c0 + c( ) + c( ) + + c( ) + = 0 I which the ceter d the coefficiets c0, c,, c, re costts. Cosider the geometric series, Similrly, we c show tht r, it coverges if = 0 = 0 r <, with sum = d coverges for < r. So Whe cosiderig power series i geerl, questio we must swer is: For wht vlues of does the power series coverge? To fid the vlues of cosider d use the Rtio Test (or the Root Test).

22 Emple: 5 = 5! Emple: ( ) = 0 ( ) Emple: = Theorem9,3: Covergece of Power Series A power series c ( ) cetered t coverges oe of three wys: = 0. The series coverges bsolutely for ll i which cse the itervl of covergece is, d the rdius of covergece is R =.. There is rel umber R > 0 such tht the series coverges bsolutely for < R d diverges for > R, i which cse the rdius of covergece is R. (Chec the edpoits.). 3. The series coverges oly t, i which cse the itervl of covergece is R = 0. Note R is clled the rdius of covergece of the power series. The itervl of rdius R cetered t = is clled the itervl of covergece R, + R, R, + R, R, + R, R, + R { ( ] [ ) [ ]} ( + ) Emple: Fid the rdius d itervl of covergece for = 0 +

23 Theorem 9.4 Combiig Power Series Suppose the power series c d respectively, o the itervl I. d coverge bsolutely to f d g,. Sum d Differece: The power series ( c d ) ± coverges bsolutely to f ± g o I.. Multiplictio by power: The power series coverges bsolutely to c = c + m m m f o I, provided m is iteger such tht m 0 h 3. Compositio: If m + for ll terms i the series. = b, where m is positive iteger d b is rel umber, the power series c h coverges bsolutely tot eh composite fuctio tht h( ) is i I. f h for ll such Emple: Give the power series for for:. + = d coverges for < fid the power series = Theorem 9.5 Differetitig d Itegrtig Power Series Let the fuctio f be defied by the power series c o its itervl of covergece I.. f is cotiuous fuctio o I.. The power series my be differetited or itegrted term by term, d the resultig power series coverges to f '( ) or f d + C, respectively, t ll poits i the iterior of I, where C is rbitrry costt. 3

24 Emple: Recll tht Fid f '. ( ) 3 4 = 0 = = +. f "( ) 3. d + Emple: The series Coverges to e for ll. d e d. Fid the series for e = ! 3! 4! 5!. Do you get the series for e? Epli your swer. b. Fid the series for e d. Do you get the series for e? Epli your swer. c. Replce by i the series for multiply the series for e d e e for ll. The to fid the first si terms of series for e e. e to fid the series tht coverges to 4

25 Sectio 9.3 Tylor Series Defiitio Tylor/Mcluri Series for Fuctio Suppose the fuctio f hs derivtives of ll orders o itervl cotiig the poit. The Tylor Series geerted by f t = is ( f ) f = ( )! = 0 ( 3 ) ( ) 3 f " f f = f + f '( ) ! 3!! The Mcluri Series is the Tylor series geerted by f t = 0 ( f ) ( 0) f =! = 0 ( 3 ) ( ) f "0 f 0 3 f 0 = f ( 0 ) + f ' ! 3!! Emple: Fid the Mcluri series for f = + Emple: Fid the Mcluri series for f = cos Emple: Fid the Mcluri series for f = e 5

26 Emple: Fid the Tylor series of f = for =. Emple: Fid the Tylor series of f = cos t = π. Theorem 9.7 Covergece of Tylor Series Let f hve derivtives of ll orders o ope itervl I cotiig. The Tylor series for f lim R = 0 for ll i I where cetered t coverges to f for ll i I if d oly if ( + f ) ( c) ( +! ) R = Is the remider t (with c betwee d ). + 6

27 Sectio 9.4 Worig with Tylor Series Applyig Tylor Series Tylor series c dded subtrcted, d multiplied by costts d the resultig series is Tylor series with itervl of coverges the itersectio of their respective itervls of covergece. Emple: Fid the Tylor series t 0 f = e = of 5 Emple: Fid the Tylor series t 0 = of f = cos( π ) Emple: Epress e d s power series. Emple: Epress e d s power series. Emple: Use series to evlute e lim 0 e 7